Statistical Methods for Financial and other Dynamical Stochastic
Models
by Kacha Dzhaparidze and Peter Spreij
The high capacity of present day computers has enabled the use
of complex stochastic models because data on the system under
study can be obtained in huge amounts and analyzed by simulation
techniques or other numerical methods. For instance, at the stock
exchanges, time and price are recorded for every single trade.
Mathematical finance is an example of a field with a vigorous
development of new models. The development of statistical methods
for stochastic process models, however, lags behind, with the
result that far too often statistical methods have been applied
that, although they can be relatively sophisticated, suffer from
shortcomings because they do not fully take into account and exploit
the structure of the new models. Researchers at CWI aim at making
a major contribution to the theory of statistical inference for
stochastic processes.
The research is carried out in close collaboration with many researchers
in The Netherlands and elsewhere in Europe. The theoretical work
uses the methods of modern probability theory including stochastic
calculus. A more applied project objective is the statistical
analysis and modelling of financial data such as stock prices,
interest rates, exchange rates and prices of options and other
derivative assets, and the development of more realistic models
for these than those presently used in the financial industry.
There are increasing demands (including new legislation) that
banks and other financial institutions improve the management
of their risk from holding positions in securities. This will
require use of more realistic and sophisticated mathematical models
as well as improved statistical procedures to evaluate prices
of financial assets.
Mathematical finance is an example of a field where data analysis
is, in practice, very often done by means of traditional discrete
time models, whereas most of the models used for pricing derivative
assets are continuous-time models. Continuous-time models have
the additional advantage that they can be analysed by means of
the powerful tools of stochastic calculus, so that results can
often be obtained even for very complicated models. In many applications,
however, one has to take into consideration that data are obtained
at discrete time points, so inference methods for discretely observed
continuous-time processes are to be applied. In recent years,
statistical methods for discrete time observations from diffusion-type
processes has started to attract attention and it appears that
there are many challenging mathematical problems involved. A survey
paper on this subject by Dzhaparidze, Spreij and Van Zanten will
soon appear in Statistica Neerlandica.
Very often the complexity of the models in question prevents exact
calculation of the statistical properties of the methods developed.
An example is calculation of the variances of estimators that
are often used to choose the most efficient member of a family
of estimators. Computer simulations are then a useful tool, but
it is important to have a mathematical theory with which simulation
results can be compared. Asymptotic statistical theory can play
this role, being therefore an important research objective at
CWI. In recent years Dzhaparidze and Spreij have published a number
of papers on parameter estimation problems in a general context
of semimartingales.
Asymptotic methods can also be used to approximate complex models
by simpler ones for inferential purposes. Moreover, the theory
of asymptotic equivalence of experiments will be used to simplify
decision problems for complex stochastic models to those of Gaussian
or Poisson models that approximate them in the deficiency distance.
This method can also be used to the approximation of discrete-time
models by continuous time-models. Certain rudimentary ideas and
facts on the relationship between these models has been reported
by Dzhaparidze in a series of three papers in CWI Quarterly. These
papers gave rise to a textbook on options valuation which is recently
completed and intended for publication at CWI.
The research described above will be further developed in close
collaboration with research teams in, eg, Paris, Berlin, Copenhagen,
Freiburg, Helsinki and Padova. Most of these teams have been involved
in the HCM research programme ‘Statistical Inference for Stochastic
Processes’. Contacts between the members of these teams are currently
maintained or reinforced at annual workshops, recently in Munzingen
(Freiburg). The collaboration with E. Valkeila (Helsinki), in
particular, proved to be quite fruitful. A number of joint papers
on general parametric families of statistical experiments were
published, and others are scheduled for this year.
Please contact:
Kacha Dzhaparidze - CWI
Tel: +31 20 592 4089
E-mail: kacha@cwi.nl
